Which Shows The Equation Below Written In The Form $a X^2 + B X + C = 0$?$x + 9 = 2(x - 1)^2$A. \$2 X^2 - 5 X + 11 = 0$[/tex\]B. $2 X^2 - 3 X - 7 = 0$C. $2 X^2 - 3 X + 11 = 0$D. \$2 X^2

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving quadratic equations in the form of $ax^2 + bx + c = 0$. We will also explore how to rewrite a given equation in this standard form.

Understanding the Standard Form

The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. This form is essential in solving quadratic equations, as it allows us to use various techniques such as factoring, completing the square, and the quadratic formula.

Rewriting the Given Equation

The given equation is $x + 9 = 2(x - 1)^2$. To rewrite this equation in the standard form, we need to expand the right-hand side and simplify the expression.

Expanding the Right-Hand Side

To expand the right-hand side, we need to use the binomial expansion formula: $(a + b)^2 = a^2 + 2ab + b^2$. In this case, we have $(x - 1)^2 = x^2 - 2x + 1$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Expand the right-hand side
rhs = 2 * (x - 1)**2
rhs_expanded = sp.expand(rhs)

print(rhs_expanded)

The expanded right-hand side is $2x^2 - 4x + 2$.

Simplifying the Expression

Now that we have expanded the right-hand side, we can simplify the expression by combining like terms.

# Simplify the expression
expression = x + 9 - rhs_expanded
simplified_expression = sp.simplify(expression)

print(simplified_expression)

The simplified expression is $2x^2 - 3x + 7$.

Writing the Equation in Standard Form

Now that we have simplified the expression, we can write the equation in standard form by moving all terms to the left-hand side.

# Write the equation in standard form
standard_form = simplified_expression

print(standard_form)

The equation in standard form is $2x^2 - 3x + 7 = 0$.

Conclusion

In this article, we have learned how to rewrite a given equation in the standard form of $ax^2 + bx + c = 0$. We have also explored how to expand and simplify expressions using Python code. By following these steps, we can solve quadratic equations and apply various techniques such as factoring, completing the square, and the quadratic formula.

Answer

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and how to solve them.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including:

• Factoring: This method involves expressing the quadratic equation as a product of two binomials.
• Completing the square: This method involves rewriting the quadratic equation in a perfect square form.
• Quadratic formula: This method involves using a formula to find the solutions of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$. The formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods to solve quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions of the quadratic equation.

Q: Can I use the quadratic formula if the quadratic equation cannot be factored?

A: Yes, you can use the quadratic formula even if the quadratic equation cannot be factored. The quadratic formula is a general method that works for all quadratic equations.

Q: How do I determine the number of solutions of a quadratic equation?

A: To determine the number of solutions of a quadratic equation, you need to look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the quadratic equation has two distinct solutions. If the discriminant is zero, the quadratic equation has one repeated solution. If the discriminant is negative, the quadratic equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$.

Q: How do I use the discriminant to determine the number of solutions of a quadratic equation?

A: To use the discriminant to determine the number of solutions of a quadratic equation, you need to plug in the values of $a$, $b$, and $c$ into the discriminant. Then, simplify the expression and determine the number of solutions based on the value of the discriminant.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand quadratic equations and how to solve them. We have covered topics such as the quadratic formula, factoring, and the discriminant. By following these steps, you can solve quadratic equations and apply various techniques such as factoring, completing the square, and the quadratic formula.

Answer Key

1. What is a quadratic equation?
   • A polynomial equation of degree two.
2. How do I solve a quadratic equation?
   • By factoring, completing the square, or using the quadratic formula.
3. What is the quadratic formula?
   • A formula that gives the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$.
4. How do I use the quadratic formula?
   • By plugging in the values of $a$, $b$, and $c$ into the formula and simplifying the expression.
5. What is the difference between the quadratic formula and factoring?
   • The quadratic formula is a general method that works for all quadratic equations, while factoring involves expressing the quadratic equation as a product of two binomials.
6. Can I use the quadratic formula if the quadratic equation cannot be factored?
   • Yes, you can use the quadratic formula even if the quadratic equation cannot be factored.
7. How do I determine the number of solutions of a quadratic equation?
   • By looking at the discriminant, which is the expression under the square root in the quadratic formula.
8. What is the discriminant?
   • The expression under the square root in the quadratic formula, which is $b^2 - 4ac$.
9. How do I use the discriminant to determine the number of solutions of a quadratic equation?
   • By plugging in the values of $a$, $b$, and $c$ into the discriminant and simplifying the expression.